This is a clarification of the previous post showing gravitational force without need of force carrier.
An instrumentalist definition of a certain quantity like mass depends on how the quantity is measured by a certain specified instrument.
According to the SI 2019 system of units, mass $m$ is determined by a Kibble balance from gravitational force $mg$ measured as electromagnetic force, where $g$ is the local gravitational constant.
We conclude that according to SI 2019 mass is gravitational mass as reaction to gravitational force. In other words, gravitational force as gradient of a gravitational potential is primary from which mass is derived as secondary as measured by a Kibble balance.
This suggests that the relation between mass density $\rho (x)$ and gravitational potential $\phi (x)$ should be viewed to have the form, with $x$ a Euclidean 3d space coordinate and $t$ a time coordinate:
- $\rho (x,t) = \Delta\phi (x,t)$ (1)
with $\phi (x)$ given and $\rho (x)$ derived by application of the differential operator $\Delta$ acting locally, rather than the standard form:
- $\Delta\phi (x,t)=\rho (x,t)$, (2)
with $\rho (x)$ given and $\phi (x)$ derived as solution to Poisson's equation as a differential equation.
We can view (1) as local instant action by differentiation, while (2) requires instant action at distance as integration/summation.
By adopting (1) as the true connection between gravitational mass and gravitational potential, which fits with SI 2019 and is explored in
many posts, we can thus circumvent to need of
instant action at distance, which has been viewed as a stumbling stone for Newtonian mechanics, motivating Einstein's relativistic mechanics coming with many new difficulties.
Adopting (1) we thus (i) adhere to SI 2019, (ii) circumvent instant action at distance and (iii) do not need Einstein, as a hat trick.
In Newtonian mechanics inertial mass is defined as gravitational mass and their equivalence is not an assumption or Principle as in Einstein's mechanics.
In the Standard Model, mass is not defined as gravitational mass as in SI 2019, but as an intrinsic quality/property carried by matter, which is formed in a very strange way by the Higg's mechanism.
Altogether, mass as gravitational mass, appears as a very useful concept. It makes sense to define mass as gravitational mass, because gravitation is present virtually everywhere. A special feature is that free fall as motion under gravitational force, without presence of forces of other nature, is independent of the size of the mass, which defines mass as a form of sensitivity to gravitation which is additive: The mass of two particles of the same kind is twice that of one particle.
But this is not how a modern physicist approaches the problem of defining what mass is and how to measure it. A modern physicist is trained to believe that mass according to the Standard Model is defined by QFT through the Higg's mechanism, and then connected to gravitation by General Relativity GR. The trouble is the QFT and GR are incompatible and so the mass problem is a mess problem.